Abstract

Dielectric particles supporting both magnetic and electric Mie resonances are shown to be able to either reflect or collect the light emitted by a single photon source. An analytical model accurately predicts the scattering behavior of a single dielectric particle electromagnetically coupled to the electric dipole transition moment of a quantum emitter. We derive near field extensions of the Kerker conditions in order to determine the conditions that strongly reduce scattering in either the forward or backward directions. This concept is then employed to design a lossless dielectric collector element whose directivity is boosted by the coherent scattering of both electric and magnetic dipoles.

© 2012 OSA

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    [CrossRef] [PubMed]
  35. M. P. Busson, B. Rolly, B. Stout, N. Bonod, and S. Bidault, “Accelerated single photon emission from dye molecule driven nanoantennas assembled on DNA,” Nat. Commun.3, 962 (2012).
    [CrossRef] [PubMed]
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    [CrossRef]
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    [CrossRef]
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2012 (6)

W. Ahn, S. V. Boriskina, Y. Hong, and B. M. Reinhard, “Photonic–plasmonic mode coupling in on-chip integrated optoplasmonic molecules,” ACS Nano6, 951–960 (2012).
[CrossRef]

A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett.12, 3749–3755 (2012).
[CrossRef] [PubMed]

W. Liu, A. E. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Broadband unidirectional scattering by magneto-electric core–shell nanoparticles,” ACS Nano6, 5489–5497 (2012).
[CrossRef] [PubMed]

B. Rolly, B. Bebey, S. Bidault, B. Stout, and N. Bonod, “Promoting magnetic dipolar transition in trivalent lanthanide ions with lossless Mie resonances,” Phys. Rev. B85, 245432 (2012).
[CrossRef]

H. Wang, X. Li, A. Pyatenko, and N. Koshizaki, “Gallium phosphide spherical particles by pulsed laser irradiation in liquid,” Sci. Adv. Mater.4, 544–547 (2012).
[CrossRef]

M. P. Busson, B. Rolly, B. Stout, N. Bonod, and S. Bidault, “Accelerated single photon emission from dye molecule driven nanoantennas assembled on DNA,” Nat. Commun.3, 962 (2012).
[CrossRef] [PubMed]

2011 (12)

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics5, 83–90 (2011).
[CrossRef]

A. Krasnok, A. Miroshnichenko, P. Belov, and Y. Kivshar, “Huygens optical elements and Yagi-Uda nanoantennas based on dielectric nanoparticles,” JETP Lett.94, 593–598 (2011).
[CrossRef]

D. S. Filonov, A. E. Krasnok, A. P. Slobozhanyuk, P. V. Kapitanova, E. A. Nenasheva, Y. S. Kivshar, and P. A. Belov, “Experimental verification of the concept of all-dielectric nanoantennas,” Appl. Phys. Lett.100, 201113 (2012).

B. Rolly, B. Stout, and N. Bonod, “Metallic dimers: When bonding transverse modes shine light,” Phys. Rev. B84, 125420 (2011).
[CrossRef]

M. P. Busson, B. Rolly, B. Stout, N. Bonod, E. Larquet, A. Polman, and S. Bidault, “Optical and topological characterization of gold nanoparticle dimers linked by a single DNA double strand,” Nano Lett.11, 5060–5065 (2011).
[CrossRef] [PubMed]

B. Rolly, B. Stout, S. Bidault, and N. Bonod, “Crucial role of the emitter–particle distance on the directivity of optical antennas,” Opt. Lett.36, 3368–3370 (2011).
[CrossRef] [PubMed]

M. Nieto-Vesperinas, R. Gomez-Medina, and J. J. Saenz, “Angle-suppressed scattering and optical forces on submicrometer dielectric particles,” J. Opt. Soc. Am. A28, 54–60 (2011).
[CrossRef]

R. Gomez-Medina, B. Garcia-Camara, I. Suarez-Lacalle, F. Gonzalez, F. Moreno, M. Nieto-Vesperinas, and J. J. Saenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophotonics5, 053512 (2011).
[CrossRef]

R. Paniagua-Domínguez, F. López-Tejeira, R. Marqués, and J. A. Sánchez-Gil, “Metallo-dielectric core–shell nanospheres as building blocks for optical three-dimensional isotropic negative-index metamaterials,” New J. Phys.13, 123017 (2011).
[CrossRef]

A. García-Etxarri, R. Gómez-Medina, L. S. Froufe-Pérez, C. López, L. Chantada, F. Scheffold, J. Aizpurua, M. Nieto-Vesperinas, and J. J. Sáenz, “Strong magnetic response of submicron silicon particles in the infrared,” Opt. Express19, 4815–4826 (2011).
[CrossRef] [PubMed]

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Goetzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics5, 166–169 (2011).
[CrossRef]

B. Stout, A. Devilez, B. Rolly, and N. Bonod, “Multipole methods for nanoantennas design: applications to Yagi-Uda configurations,” J. Opt. Soc. Am. B28, 1213–1223 (2011).
[CrossRef]

2010 (5)

A. Devilez, B. Stout, and N. Bonod, “Compact metallo-dielectric optical antenna for ultra directional and enhanced radiative emission,” ACS Nano4, 3390–3396 (2010).
[CrossRef] [PubMed]

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science329, 930–933 (2010).
[CrossRef] [PubMed]

R. Esteban, T. V. Teperik, and J. J. Greffet, “Optical patch antennas for single photon emission using surface plasmon resonances,” Phys. Rev. Lett.104, 026802 (2010).
[CrossRef] [PubMed]

N. Bonod, A. Devilez, B. Rolly, S. Bidault, and B. Stout, “Ultracompact and unidirectional metallic antennas,” Phys. Rev. B82, 115429 (2010).
[CrossRef]

A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B82, 045404 (2010).
[CrossRef]

2009 (5)

T. Pakizeh and M. Kall, “Unidirectional ultracompact optical antennas,” Nano Lett.9, 2343–2349 (2009).
[CrossRef] [PubMed]

A. Koenderink, “Plasmon nanoparticle array waveguides for single photon and single plasmon sources,” Nano Lett.9, 4228–4233 (2009).
[CrossRef] [PubMed]

D. Gérard, A. Devilez, H. Aouani, B. Stout, N. Bonod, J. Wenger, E. Popov, and H. Rigneault, “Efficient excitation and collection of single-molecule fluorescence close to a dielectric microsphere,” J. Opt. Soc. Am. B26, 1473– 1478 (2009).
[CrossRef]

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today12, 60–69 (2009).
[CrossRef]

K. Vynck, D. Felbacq, E. Centeno, A. I. Căbuz, D. Cassagne, and B. Guizal, “All-dielectric rod-type metamaterials at optical frequencies,” Phys. Rev. Lett.102, 133901 (2009).
[CrossRef] [PubMed]

2008 (2)

2007 (4)

J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: A Yagi-Yda nanoantenna in the optical domain,” Phys. Rev. B76, 245403 (2007).
[CrossRef]

H. F. Hofman, T. Kosako, and Y. Kadoya, “Design parameters for a nano-optical Yagi-Uda antenna,” New J. Phys.9, 217 (2007).
[CrossRef]

P. Bharadwaj and L. Novotny, “Spectral dependence of single molecule fluorescence enhancement,” Opt. Express15, 14266–14274 (2007).
[CrossRef] [PubMed]

J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett.99, 107401 (2007).
[CrossRef] [PubMed]

2006 (1)

R. Carminati, J. J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun.261, 368–375 (2006).
[CrossRef]

2005 (1)

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B72, 193103 (2005).
[CrossRef]

1983 (1)

Ahn, W.

W. Ahn, S. V. Boriskina, Y. Hong, and B. M. Reinhard, “Photonic–plasmonic mode coupling in on-chip integrated optoplasmonic molecules,” ACS Nano6, 951–960 (2012).
[CrossRef]

Aitchison, J. S.

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B72, 193103 (2005).
[CrossRef]

Aizpurua, J.

Aouani, H.

Bebey, B.

B. Rolly, B. Bebey, S. Bidault, B. Stout, and N. Bonod, “Promoting magnetic dipolar transition in trivalent lanthanide ions with lossless Mie resonances,” Phys. Rev. B85, 245432 (2012).
[CrossRef]

Belov, P.

A. Krasnok, A. Miroshnichenko, P. Belov, and Y. Kivshar, “Huygens optical elements and Yagi-Uda nanoantennas based on dielectric nanoparticles,” JETP Lett.94, 593–598 (2011).
[CrossRef]

Belov, P. A.

D. S. Filonov, A. E. Krasnok, A. P. Slobozhanyuk, P. V. Kapitanova, E. A. Nenasheva, Y. S. Kivshar, and P. A. Belov, “Experimental verification of the concept of all-dielectric nanoantennas,” Appl. Phys. Lett.100, 201113 (2012).

Bharadwaj, P.

Bidault, S.

B. Rolly, B. Bebey, S. Bidault, B. Stout, and N. Bonod, “Promoting magnetic dipolar transition in trivalent lanthanide ions with lossless Mie resonances,” Phys. Rev. B85, 245432 (2012).
[CrossRef]

M. P. Busson, B. Rolly, B. Stout, N. Bonod, and S. Bidault, “Accelerated single photon emission from dye molecule driven nanoantennas assembled on DNA,” Nat. Commun.3, 962 (2012).
[CrossRef] [PubMed]

M. P. Busson, B. Rolly, B. Stout, N. Bonod, E. Larquet, A. Polman, and S. Bidault, “Optical and topological characterization of gold nanoparticle dimers linked by a single DNA double strand,” Nano Lett.11, 5060–5065 (2011).
[CrossRef] [PubMed]

B. Rolly, B. Stout, S. Bidault, and N. Bonod, “Crucial role of the emitter–particle distance on the directivity of optical antennas,” Opt. Lett.36, 3368–3370 (2011).
[CrossRef] [PubMed]

N. Bonod, A. Devilez, B. Rolly, S. Bidault, and B. Stout, “Ultracompact and unidirectional metallic antennas,” Phys. Rev. B82, 115429 (2010).
[CrossRef]

Bonod, N.

M. P. Busson, B. Rolly, B. Stout, N. Bonod, and S. Bidault, “Accelerated single photon emission from dye molecule driven nanoantennas assembled on DNA,” Nat. Commun.3, 962 (2012).
[CrossRef] [PubMed]

B. Rolly, B. Bebey, S. Bidault, B. Stout, and N. Bonod, “Promoting magnetic dipolar transition in trivalent lanthanide ions with lossless Mie resonances,” Phys. Rev. B85, 245432 (2012).
[CrossRef]

B. Rolly, B. Stout, and N. Bonod, “Metallic dimers: When bonding transverse modes shine light,” Phys. Rev. B84, 125420 (2011).
[CrossRef]

M. P. Busson, B. Rolly, B. Stout, N. Bonod, E. Larquet, A. Polman, and S. Bidault, “Optical and topological characterization of gold nanoparticle dimers linked by a single DNA double strand,” Nano Lett.11, 5060–5065 (2011).
[CrossRef] [PubMed]

B. Rolly, B. Stout, S. Bidault, and N. Bonod, “Crucial role of the emitter–particle distance on the directivity of optical antennas,” Opt. Lett.36, 3368–3370 (2011).
[CrossRef] [PubMed]

B. Stout, A. Devilez, B. Rolly, and N. Bonod, “Multipole methods for nanoantennas design: applications to Yagi-Uda configurations,” J. Opt. Soc. Am. B28, 1213–1223 (2011).
[CrossRef]

A. Devilez, B. Stout, and N. Bonod, “Compact metallo-dielectric optical antenna for ultra directional and enhanced radiative emission,” ACS Nano4, 3390–3396 (2010).
[CrossRef] [PubMed]

N. Bonod, A. Devilez, B. Rolly, S. Bidault, and B. Stout, “Ultracompact and unidirectional metallic antennas,” Phys. Rev. B82, 115429 (2010).
[CrossRef]

D. Gérard, A. Devilez, H. Aouani, B. Stout, N. Bonod, J. Wenger, E. Popov, and H. Rigneault, “Efficient excitation and collection of single-molecule fluorescence close to a dielectric microsphere,” J. Opt. Soc. Am. B26, 1473– 1478 (2009).
[CrossRef]

Boriskina, S. V.

W. Ahn, S. V. Boriskina, Y. Hong, and B. M. Reinhard, “Photonic–plasmonic mode coupling in on-chip integrated optoplasmonic molecules,” ACS Nano6, 951–960 (2012).
[CrossRef]

Bouhelier, A.

Bozhevolnyi, S. I.

A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett.12, 3749–3755 (2012).
[CrossRef] [PubMed]

Brongersma, M. L.

J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett.99, 107401 (2007).
[CrossRef] [PubMed]

Busson, M. P.

M. P. Busson, B. Rolly, B. Stout, N. Bonod, and S. Bidault, “Accelerated single photon emission from dye molecule driven nanoantennas assembled on DNA,” Nat. Commun.3, 962 (2012).
[CrossRef] [PubMed]

M. P. Busson, B. Rolly, B. Stout, N. Bonod, E. Larquet, A. Polman, and S. Bidault, “Optical and topological characterization of gold nanoparticle dimers linked by a single DNA double strand,” Nano Lett.11, 5060–5065 (2011).
[CrossRef] [PubMed]

Cabuz, A. I.

K. Vynck, D. Felbacq, E. Centeno, A. I. Căbuz, D. Cassagne, and B. Guizal, “All-dielectric rod-type metamaterials at optical frequencies,” Phys. Rev. Lett.102, 133901 (2009).
[CrossRef] [PubMed]

Carminati, R.

R. Carminati, J. J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun.261, 368–375 (2006).
[CrossRef]

Cassagne, D.

K. Vynck, D. Felbacq, E. Centeno, A. I. Căbuz, D. Cassagne, and B. Guizal, “All-dielectric rod-type metamaterials at optical frequencies,” Phys. Rev. Lett.102, 133901 (2009).
[CrossRef] [PubMed]

Centeno, E.

K. Vynck, D. Felbacq, E. Centeno, A. I. Căbuz, D. Cassagne, and B. Guizal, “All-dielectric rod-type metamaterials at optical frequencies,” Phys. Rev. Lett.102, 133901 (2009).
[CrossRef] [PubMed]

Chantada, L.

Chen, X. W.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Goetzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics5, 166–169 (2011).
[CrossRef]

Chichkov, B. N.

A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett.12, 3749–3755 (2012).
[CrossRef] [PubMed]

A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B82, 045404 (2010).
[CrossRef]

Curto, A. G.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science329, 930–933 (2010).
[CrossRef] [PubMed]

Dereux, A.

des Francs, G. C.

Devilez, A.

B. Stout, A. Devilez, B. Rolly, and N. Bonod, “Multipole methods for nanoantennas design: applications to Yagi-Uda configurations,” J. Opt. Soc. Am. B28, 1213–1223 (2011).
[CrossRef]

N. Bonod, A. Devilez, B. Rolly, S. Bidault, and B. Stout, “Ultracompact and unidirectional metallic antennas,” Phys. Rev. B82, 115429 (2010).
[CrossRef]

A. Devilez, B. Stout, and N. Bonod, “Compact metallo-dielectric optical antenna for ultra directional and enhanced radiative emission,” ACS Nano4, 3390–3396 (2010).
[CrossRef] [PubMed]

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J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: A Yagi-Yda nanoantenna in the optical domain,” Phys. Rev. B76, 245403 (2007).
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A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett.12, 3749–3755 (2012).
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R. Esteban, T. V. Teperik, and J. J. Greffet, “Optical patch antennas for single photon emission using surface plasmon resonances,” Phys. Rev. Lett.104, 026802 (2010).
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A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett.12, 3749–3755 (2012).
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A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B82, 045404 (2010).
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K. Vynck, D. Felbacq, E. Centeno, A. I. Căbuz, D. Cassagne, and B. Guizal, “All-dielectric rod-type metamaterials at optical frequencies,” Phys. Rev. Lett.102, 133901 (2009).
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D. S. Filonov, A. E. Krasnok, A. P. Slobozhanyuk, P. V. Kapitanova, E. A. Nenasheva, Y. S. Kivshar, and P. A. Belov, “Experimental verification of the concept of all-dielectric nanoantennas,” Appl. Phys. Lett.100, 201113 (2012).

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R. Gomez-Medina, B. Garcia-Camara, I. Suarez-Lacalle, F. Gonzalez, F. Moreno, M. Nieto-Vesperinas, and J. J. Saenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophotonics5, 053512 (2011).
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K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Goetzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics5, 166–169 (2011).
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R. Gomez-Medina, B. Garcia-Camara, I. Suarez-Lacalle, F. Gonzalez, F. Moreno, M. Nieto-Vesperinas, and J. J. Saenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophotonics5, 053512 (2011).
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M. Nieto-Vesperinas, R. Gomez-Medina, and J. J. Saenz, “Angle-suppressed scattering and optical forces on submicrometer dielectric particles,” J. Opt. Soc. Am. A28, 54–60 (2011).
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Gonzalez, F.

R. Gomez-Medina, B. Garcia-Camara, I. Suarez-Lacalle, F. Gonzalez, F. Moreno, M. Nieto-Vesperinas, and J. J. Saenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophotonics5, 053512 (2011).
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R. Esteban, T. V. Teperik, and J. J. Greffet, “Optical patch antennas for single photon emission using surface plasmon resonances,” Phys. Rev. Lett.104, 026802 (2010).
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R. Carminati, J. J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun.261, 368–375 (2006).
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K. Vynck, D. Felbacq, E. Centeno, A. I. Căbuz, D. Cassagne, and B. Guizal, “All-dielectric rod-type metamaterials at optical frequencies,” Phys. Rev. Lett.102, 133901 (2009).
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R. Carminati, J. J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun.261, 368–375 (2006).
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H. F. Hofman, T. Kosako, and Y. Kadoya, “Design parameters for a nano-optical Yagi-Uda antenna,” New J. Phys.9, 217 (2007).
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W. Ahn, S. V. Boriskina, Y. Hong, and B. M. Reinhard, “Photonic–plasmonic mode coupling in on-chip integrated optoplasmonic molecules,” ACS Nano6, 951–960 (2012).
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H. F. Hofman, T. Kosako, and Y. Kadoya, “Design parameters for a nano-optical Yagi-Uda antenna,” New J. Phys.9, 217 (2007).
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T. Pakizeh and M. Kall, “Unidirectional ultracompact optical antennas,” Nano Lett.9, 2343–2349 (2009).
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D. S. Filonov, A. E. Krasnok, A. P. Slobozhanyuk, P. V. Kapitanova, E. A. Nenasheva, Y. S. Kivshar, and P. A. Belov, “Experimental verification of the concept of all-dielectric nanoantennas,” Appl. Phys. Lett.100, 201113 (2012).

Kerker, M.

Kivshar, Y.

A. Krasnok, A. Miroshnichenko, P. Belov, and Y. Kivshar, “Huygens optical elements and Yagi-Uda nanoantennas based on dielectric nanoparticles,” JETP Lett.94, 593–598 (2011).
[CrossRef]

Kivshar, Y. S.

W. Liu, A. E. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Broadband unidirectional scattering by magneto-electric core–shell nanoparticles,” ACS Nano6, 5489–5497 (2012).
[CrossRef] [PubMed]

D. S. Filonov, A. E. Krasnok, A. P. Slobozhanyuk, P. V. Kapitanova, E. A. Nenasheva, Y. S. Kivshar, and P. A. Belov, “Experimental verification of the concept of all-dielectric nanoantennas,” Appl. Phys. Lett.100, 201113 (2012).

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H. F. Hofman, T. Kosako, and Y. Kadoya, “Design parameters for a nano-optical Yagi-Uda antenna,” New J. Phys.9, 217 (2007).
[CrossRef]

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H. Wang, X. Li, A. Pyatenko, and N. Koshizaki, “Gallium phosphide spherical particles by pulsed laser irradiation in liquid,” Sci. Adv. Mater.4, 544–547 (2012).
[CrossRef]

Krasnok, A.

A. Krasnok, A. Miroshnichenko, P. Belov, and Y. Kivshar, “Huygens optical elements and Yagi-Uda nanoantennas based on dielectric nanoparticles,” JETP Lett.94, 593–598 (2011).
[CrossRef]

Krasnok, A. E.

D. S. Filonov, A. E. Krasnok, A. P. Slobozhanyuk, P. V. Kapitanova, E. A. Nenasheva, Y. S. Kivshar, and P. A. Belov, “Experimental verification of the concept of all-dielectric nanoantennas,” Appl. Phys. Lett.100, 201113 (2012).

Kreuzer, M. P.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science329, 930–933 (2010).
[CrossRef] [PubMed]

Kukura, P.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Goetzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics5, 166–169 (2011).
[CrossRef]

Larquet, E.

M. P. Busson, B. Rolly, B. Stout, N. Bonod, E. Larquet, A. Polman, and S. Bidault, “Optical and topological characterization of gold nanoparticle dimers linked by a single DNA double strand,” Nano Lett.11, 5060–5065 (2011).
[CrossRef] [PubMed]

Lee, K. G.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Goetzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics5, 166–169 (2011).
[CrossRef]

Lettow, R.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Goetzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics5, 166–169 (2011).
[CrossRef]

Li, J.

J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: A Yagi-Yda nanoantenna in the optical domain,” Phys. Rev. B76, 245403 (2007).
[CrossRef]

Li, X.

H. Wang, X. Li, A. Pyatenko, and N. Koshizaki, “Gallium phosphide spherical particles by pulsed laser irradiation in liquid,” Sci. Adv. Mater.4, 544–547 (2012).
[CrossRef]

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Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today12, 60–69 (2009).
[CrossRef]

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W. Liu, A. E. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Broadband unidirectional scattering by magneto-electric core–shell nanoparticles,” ACS Nano6, 5489–5497 (2012).
[CrossRef] [PubMed]

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López-Tejeira, F.

R. Paniagua-Domínguez, F. López-Tejeira, R. Marqués, and J. A. Sánchez-Gil, “Metallo-dielectric core–shell nanospheres as building blocks for optical three-dimensional isotropic negative-index metamaterials,” New J. Phys.13, 123017 (2011).
[CrossRef]

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A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B82, 045404 (2010).
[CrossRef]

Marqués, R.

R. Paniagua-Domínguez, F. López-Tejeira, R. Marqués, and J. A. Sánchez-Gil, “Metallo-dielectric core–shell nanospheres as building blocks for optical three-dimensional isotropic negative-index metamaterials,” New J. Phys.13, 123017 (2011).
[CrossRef]

Miroshnichenko, A.

A. Krasnok, A. Miroshnichenko, P. Belov, and Y. Kivshar, “Huygens optical elements and Yagi-Uda nanoantennas based on dielectric nanoparticles,” JETP Lett.94, 593–598 (2011).
[CrossRef]

Miroshnichenko, A. E.

W. Liu, A. E. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Broadband unidirectional scattering by magneto-electric core–shell nanoparticles,” ACS Nano6, 5489–5497 (2012).
[CrossRef] [PubMed]

Mojahedi, M.

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B72, 193103 (2005).
[CrossRef]

Moreno, F.

R. Gomez-Medina, B. Garcia-Camara, I. Suarez-Lacalle, F. Gonzalez, F. Moreno, M. Nieto-Vesperinas, and J. J. Saenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophotonics5, 053512 (2011).
[CrossRef]

Nenasheva, E. A.

D. S. Filonov, A. E. Krasnok, A. P. Slobozhanyuk, P. V. Kapitanova, E. A. Nenasheva, Y. S. Kivshar, and P. A. Belov, “Experimental verification of the concept of all-dielectric nanoantennas,” Appl. Phys. Lett.100, 201113 (2012).

Neshev, D. N.

W. Liu, A. E. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Broadband unidirectional scattering by magneto-electric core–shell nanoparticles,” ACS Nano6, 5489–5497 (2012).
[CrossRef] [PubMed]

Nieto-Vesperinas, M.

Novikov, S. M.

A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett.12, 3749–3755 (2012).
[CrossRef] [PubMed]

Novotny, L.

Pakizeh, T.

T. Pakizeh and M. Kall, “Unidirectional ultracompact optical antennas,” Nano Lett.9, 2343–2349 (2009).
[CrossRef] [PubMed]

Paniagua-Domínguez, R.

R. Paniagua-Domínguez, F. López-Tejeira, R. Marqués, and J. A. Sánchez-Gil, “Metallo-dielectric core–shell nanospheres as building blocks for optical three-dimensional isotropic negative-index metamaterials,” New J. Phys.13, 123017 (2011).
[CrossRef]

Polman, A.

M. P. Busson, B. Rolly, B. Stout, N. Bonod, E. Larquet, A. Polman, and S. Bidault, “Optical and topological characterization of gold nanoparticle dimers linked by a single DNA double strand,” Nano Lett.11, 5060–5065 (2011).
[CrossRef] [PubMed]

Popov, E.

Pyatenko, A.

H. Wang, X. Li, A. Pyatenko, and N. Koshizaki, “Gallium phosphide spherical particles by pulsed laser irradiation in liquid,” Sci. Adv. Mater.4, 544–547 (2012).
[CrossRef]

Quidant, R.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science329, 930–933 (2010).
[CrossRef] [PubMed]

Reinhard, B. M.

W. Ahn, S. V. Boriskina, Y. Hong, and B. M. Reinhard, “Photonic–plasmonic mode coupling in on-chip integrated optoplasmonic molecules,” ACS Nano6, 951–960 (2012).
[CrossRef]

Reinhardt, C.

A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett.12, 3749–3755 (2012).
[CrossRef] [PubMed]

A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B82, 045404 (2010).
[CrossRef]

Renn, A.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Goetzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics5, 166–169 (2011).
[CrossRef]

Rigneault, H.

Rolly, B.

B. Rolly, B. Bebey, S. Bidault, B. Stout, and N. Bonod, “Promoting magnetic dipolar transition in trivalent lanthanide ions with lossless Mie resonances,” Phys. Rev. B85, 245432 (2012).
[CrossRef]

M. P. Busson, B. Rolly, B. Stout, N. Bonod, and S. Bidault, “Accelerated single photon emission from dye molecule driven nanoantennas assembled on DNA,” Nat. Commun.3, 962 (2012).
[CrossRef] [PubMed]

B. Rolly, B. Stout, and N. Bonod, “Metallic dimers: When bonding transverse modes shine light,” Phys. Rev. B84, 125420 (2011).
[CrossRef]

M. P. Busson, B. Rolly, B. Stout, N. Bonod, E. Larquet, A. Polman, and S. Bidault, “Optical and topological characterization of gold nanoparticle dimers linked by a single DNA double strand,” Nano Lett.11, 5060–5065 (2011).
[CrossRef] [PubMed]

B. Stout, A. Devilez, B. Rolly, and N. Bonod, “Multipole methods for nanoantennas design: applications to Yagi-Uda configurations,” J. Opt. Soc. Am. B28, 1213–1223 (2011).
[CrossRef]

B. Rolly, B. Stout, S. Bidault, and N. Bonod, “Crucial role of the emitter–particle distance on the directivity of optical antennas,” Opt. Lett.36, 3368–3370 (2011).
[CrossRef] [PubMed]

N. Bonod, A. Devilez, B. Rolly, S. Bidault, and B. Stout, “Ultracompact and unidirectional metallic antennas,” Phys. Rev. B82, 115429 (2010).
[CrossRef]

Saenz, J. J.

R. Gomez-Medina, B. Garcia-Camara, I. Suarez-Lacalle, F. Gonzalez, F. Moreno, M. Nieto-Vesperinas, and J. J. Saenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophotonics5, 053512 (2011).
[CrossRef]

M. Nieto-Vesperinas, R. Gomez-Medina, and J. J. Saenz, “Angle-suppressed scattering and optical forces on submicrometer dielectric particles,” J. Opt. Soc. Am. A28, 54–60 (2011).
[CrossRef]

Sáenz, J. J.

Salandrino, A.

J. Li, A. Salandrino, and N. Engheta, “Shaping light beams in the nanometer scale: A Yagi-Yda nanoantenna in the optical domain,” Phys. Rev. B76, 245403 (2007).
[CrossRef]

Sánchez-Gil, J. A.

R. Paniagua-Domínguez, F. López-Tejeira, R. Marqués, and J. A. Sánchez-Gil, “Metallo-dielectric core–shell nanospheres as building blocks for optical three-dimensional isotropic negative-index metamaterials,” New J. Phys.13, 123017 (2011).
[CrossRef]

Sandoghdar, V.

K. G. Lee, X. W. Chen, H. Eghlidi, P. Kukura, R. Lettow, A. Renn, V. Sandoghdar, and S. Goetzinger, “A planar dielectric antenna for directional single-photon emission and near-unity collection efficiency,” Nat. Photonics5, 166–169 (2011).
[CrossRef]

Scheffold, F.

Schuller, J. A.

J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett.99, 107401 (2007).
[CrossRef] [PubMed]

Seidel, A.

A. B. Evlyukhin, C. Reinhardt, A. Seidel, B. S. Luk’yanchuk, and B. N. Chichkov, “Optical response features of Si-nanoparticle arrays,” Phys. Rev. B82, 045404 (2010).
[CrossRef]

Slobozhanyuk, A. P.

D. S. Filonov, A. E. Krasnok, A. P. Slobozhanyuk, P. V. Kapitanova, E. A. Nenasheva, Y. S. Kivshar, and P. A. Belov, “Experimental verification of the concept of all-dielectric nanoantennas,” Appl. Phys. Lett.100, 201113 (2012).

Stefani, F. D.

Stout, B.

B. Rolly, B. Bebey, S. Bidault, B. Stout, and N. Bonod, “Promoting magnetic dipolar transition in trivalent lanthanide ions with lossless Mie resonances,” Phys. Rev. B85, 245432 (2012).
[CrossRef]

M. P. Busson, B. Rolly, B. Stout, N. Bonod, and S. Bidault, “Accelerated single photon emission from dye molecule driven nanoantennas assembled on DNA,” Nat. Commun.3, 962 (2012).
[CrossRef] [PubMed]

M. P. Busson, B. Rolly, B. Stout, N. Bonod, E. Larquet, A. Polman, and S. Bidault, “Optical and topological characterization of gold nanoparticle dimers linked by a single DNA double strand,” Nano Lett.11, 5060–5065 (2011).
[CrossRef] [PubMed]

B. Rolly, B. Stout, and N. Bonod, “Metallic dimers: When bonding transverse modes shine light,” Phys. Rev. B84, 125420 (2011).
[CrossRef]

B. Stout, A. Devilez, B. Rolly, and N. Bonod, “Multipole methods for nanoantennas design: applications to Yagi-Uda configurations,” J. Opt. Soc. Am. B28, 1213–1223 (2011).
[CrossRef]

B. Rolly, B. Stout, S. Bidault, and N. Bonod, “Crucial role of the emitter–particle distance on the directivity of optical antennas,” Opt. Lett.36, 3368–3370 (2011).
[CrossRef] [PubMed]

N. Bonod, A. Devilez, B. Rolly, S. Bidault, and B. Stout, “Ultracompact and unidirectional metallic antennas,” Phys. Rev. B82, 115429 (2010).
[CrossRef]

A. Devilez, B. Stout, and N. Bonod, “Compact metallo-dielectric optical antenna for ultra directional and enhanced radiative emission,” ACS Nano4, 3390–3396 (2010).
[CrossRef] [PubMed]

D. Gérard, A. Devilez, H. Aouani, B. Stout, N. Bonod, J. Wenger, E. Popov, and H. Rigneault, “Efficient excitation and collection of single-molecule fluorescence close to a dielectric microsphere,” J. Opt. Soc. Am. B26, 1473– 1478 (2009).
[CrossRef]

Suarez-Lacalle, I.

R. Gomez-Medina, B. Garcia-Camara, I. Suarez-Lacalle, F. Gonzalez, F. Moreno, M. Nieto-Vesperinas, and J. J. Saenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophotonics5, 053512 (2011).
[CrossRef]

Taminiau, T. H.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science329, 930–933 (2010).
[CrossRef] [PubMed]

T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Enhanced directional excitation and emission of single emitters by a nano-optical Yagi-Uda antenna.” Opt. Express16, 10858–10866 (2008).
[CrossRef] [PubMed]

Taubner, T.

J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett.99, 107401 (2007).
[CrossRef] [PubMed]

Teperik, T. V.

R. Esteban, T. V. Teperik, and J. J. Greffet, “Optical patch antennas for single photon emission using surface plasmon resonances,” Phys. Rev. Lett.104, 026802 (2010).
[CrossRef] [PubMed]

van Hulst, N.

L. Novotny and N. van Hulst, “Antennas for light,” Nat. Photonics5, 83–90 (2011).
[CrossRef]

van Hulst, N. F.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science329, 930–933 (2010).
[CrossRef] [PubMed]

T. H. Taminiau, F. D. Stefani, and N. F. van Hulst, “Enhanced directional excitation and emission of single emitters by a nano-optical Yagi-Uda antenna.” Opt. Express16, 10858–10866 (2008).
[CrossRef] [PubMed]

Vigoureux, J. M.

R. Carminati, J. J. Greffet, C. Henkel, and J. M. Vigoureux, “Radiative and non-radiative decay of a single molecule close to a metallic nanoparticle,” Opt. Commun.261, 368–375 (2006).
[CrossRef]

Volpe, G.

A. G. Curto, G. Volpe, T. H. Taminiau, M. P. Kreuzer, R. Quidant, and N. F. van Hulst, “Unidirectional emission of a quantum dot coupled to a nanoantenna,” Science329, 930–933 (2010).
[CrossRef] [PubMed]

Vynck, K.

K. Vynck, D. Felbacq, E. Centeno, A. I. Căbuz, D. Cassagne, and B. Guizal, “All-dielectric rod-type metamaterials at optical frequencies,” Phys. Rev. Lett.102, 133901 (2009).
[CrossRef] [PubMed]

Wang, D.-S.

Wang, H.

H. Wang, X. Li, A. Pyatenko, and N. Koshizaki, “Gallium phosphide spherical particles by pulsed laser irradiation in liquid,” Sci. Adv. Mater.4, 544–547 (2012).
[CrossRef]

Weeber, J. C.

Wenger, J.

Wheeler, M. S.

M. S. Wheeler, J. S. Aitchison, and M. Mojahedi, “Three-dimensional array of dielectric spheres with an isotropic negative permeability at infrared frequencies,” Phys. Rev. B72, 193103 (2005).
[CrossRef]

Zhang, F.

Q. Zhao, J. Zhou, F. Zhang, and D. Lippens, “Mie resonance-based dielectric metamaterials,” Mater. Today12, 60–69 (2009).
[CrossRef]

Zhao, Q.

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Zia, R.

J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett.99, 107401 (2007).
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Zywietz, U.

A. B. Evlyukhin, S. M. Novikov, U. Zywietz, R. L. Eriksen, C. Reinhardt, S. I. Bozhevolnyi, and B. N. Chichkov, “Demonstration of magnetic dipole resonances of dielectric nanospheres in the visible region,” Nano Lett.12, 3749–3755 (2012).
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A. Devilez, B. Stout, and N. Bonod, “Compact metallo-dielectric optical antenna for ultra directional and enhanced radiative emission,” ACS Nano4, 3390–3396 (2010).
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W. Ahn, S. V. Boriskina, Y. Hong, and B. M. Reinhard, “Photonic–plasmonic mode coupling in on-chip integrated optoplasmonic molecules,” ACS Nano6, 951–960 (2012).
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W. Liu, A. E. Miroshnichenko, D. N. Neshev, and Y. S. Kivshar, “Broadband unidirectional scattering by magneto-electric core–shell nanoparticles,” ACS Nano6, 5489–5497 (2012).
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Appl. Phys. Lett. (1)

D. S. Filonov, A. E. Krasnok, A. P. Slobozhanyuk, P. V. Kapitanova, E. A. Nenasheva, Y. S. Kivshar, and P. A. Belov, “Experimental verification of the concept of all-dielectric nanoantennas,” Appl. Phys. Lett.100, 201113 (2012).

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R. Gomez-Medina, B. Garcia-Camara, I. Suarez-Lacalle, F. Gonzalez, F. Moreno, M. Nieto-Vesperinas, and J. J. Saenz, “Electric and magnetic dipolar response of germanium nanospheres: interference effects, scattering anisotropy, and optical forces,” J. Nanophotonics5, 053512 (2011).
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M. P. Busson, B. Rolly, B. Stout, N. Bonod, and S. Bidault, “Accelerated single photon emission from dye molecule driven nanoantennas assembled on DNA,” Nat. Commun.3, 962 (2012).
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J. A. Schuller, R. Zia, T. Taubner, and M. L. Brongersma, “Dielectric metamaterials based on electric and magnetic resonances of silicon carbide particles,” Phys. Rev. Lett.99, 107401 (2007).
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Figures (6)

Fig. 1
Fig. 1

(a) An electric dipole emitter oriented along the x axis is coupled to a sphere placed at a varying distance along the z axis. (b–c) Fraction of the total radiated power that is collected by the sphere, i.e. emitted in the z > 0 half-space, for a GaP sphere of varying radius (ordinate) and an electric dipole placed at a varying distance (abscissa) from the surface of the sphere. The refractive index of the embedding medium is n = 1.45 and the emission wavelength in vacuum is λ = 550nm. (b) Analytical calculations performed with the dipolar model given in Eq. (11) and (c) multipolar calculations performed with the GMT method.

Fig. 2
Fig. 2

Radiation diagrams for a 85 nm radius GaP sphere behaving as (a) a collector, distance to the emitter 10 nm and (b) a reflector, distance to the emitter 100 nm. The electric dipole emitter is oriented along the x axis, the sphere is placed in the +z direction. A magnitude of 1 corresponds to the maximal power of the emitter placed in the homogeneous background. The refractive index of the embedding medium is n = 1.45 and the emission wavelength in vacuum is λ = 550nm

Fig. 3
Fig. 3

Gain in directivity in dBi as functions of both particle radius and distance of the emitter to the surface of the particle. Same parameter as in Fig. 1. With both magnetic and electric Mie resonances (a), with electric (b) and magnetic (c) Mie resonance only. The calculations are performed with GMT by considering 10 multipoles.

Fig. 4
Fig. 4

The angular coordinates system used in this demonstration. The zenith angles θi and azimuthal angles φi are defined respectively to the 3 axis, i = x, y, z. The choice i = z corresponds to the usual spherical coordinates.

Fig. 5
Fig. 5

A sphere of GaP, 90 nm in radius, is located at a distance d = 10 nm from the emitter in (a–c), and at a distance d = 100 nm in (d–f). Radiation patterns when considering: (a,d) the electric induced dipole only, (b,e) the magnetic induced dipole only, (c,f) both electric and magnetic induced dipoles.

Fig. 6
Fig. 6

(a) Schematic of a hybrid antenna with an electric dipolar emitter longitudinally coupled to a silver dimer (particles 30 nm in radius and nanogap length of 8 nm), and transversely coupled to a GaP sphere (75 nm in radius, surface 30 nm away from the emitter). (b) Radiation diagram at the vacuum wavelength λ0 = 526 nm, when the decay rate enhancement is maximal. (c) (left scale) Radiative decay rate enhancement (full black line) of the hybrid nanoantenna and (dashed line) of the metallic dimer antenna alone. (Right scale, full blue circles) Quantum efficiency of the hybrid nanoantenna. (Inset) Gain in directivity of the hybrid antenna.

Equations (28)

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E p ( r r ^ ) = e i k r 4 π ε m ε 0 r 3 [ k 2 r 2 ( r ^ × p ) × r ^ + ( 1 i k r ) ( 3 ( r ^ p ) r ^ p ) ] ,
H p ( r r ^ ) = e i k r 4 π n 0 r c k 2 ( 1 + i k r ) r ^ × p ,
E p , ff ( r r ^ ) = e i k r 4 π ε m ε 0 r k 2 sin ( θ x ) p exp ( i ϕ p ) θ ^ x ,
H p , ff ( r r ^ ) = e i k r 4 π n 0 r c k 2 sin ( θ x ) p exp ( i ϕ p ) φ ^ x .
E p , ff ( r r ^ , d ) = e i k r 4 π ε m ε 0 r k 2 sin ( θ x ) p exp ( i ϕ p ) exp ( i k d cos θ z ) θ ^ x ,
H p , ff ( r r ^ , d ) = e i k r 4 π n 0 r c k 2 sin ( θ x ) p exp ( i ϕ p ) exp ( i k d cos θ z ) φ ^ x .
E m ( r r ^ ) = e i k r 4 π ε 0 n 0 c r k 2 ( 1 + i k r ) r ^ × m ,
H m ( r r ^ ) = e i k r 4 π r 3 [ k 2 r 2 ( r ^ × m ) × r ^ + ( 1 i k r ) ( 3 ( r ^ m ) r ^ m ) ] ,
E m , ff ( r r ^ ) = e i k r 4 π ε 0 n 0 c r k 2 sin ( θ y ) m exp ( i ϕ m ) φ ^ y ,
H m , ff ( r r ^ ) = e i k r 4 π r k 2 sin ( θ y ) m exp ( i ϕ m ) θ ^ y .
E m , ff ( r r ^ , d ) = e i k r 4 π ε 0 n 0 c r k 2 sin ( θ y ) m exp ( i ϕ m ) exp ( i k d cos θ z ) φ ^ y ,
H m , ff ( r r ^ , d ) = e i k r 4 π r k 2 sin ( θ y ) m exp ( i ϕ m ) exp ( i k d cos θ z ) θ ^ y .
E 0 ( d z ^ ) = e i k d 4 π ε m ε 0 d 3 ( 1 i k d k 2 d 2 ) x ^ ,
H 0 ( d z ^ ) = e i k d 4 π n 0 d 3 c ( k 2 d 2 + i k d ) y ^ .
α ˜ = i 3 2 k 3 a 3 a 1 ; β ˜ = i 3 2 k 3 a 3 b 1 , p in = 4 π a 3 ε 0 ε m α ˜ E 0 ( d z ^ ) ; m in = 4 π a 3 β ˜ H 0 ( d z ^ ) .
p in = 4 π a 3 ε 0 ε m α ˜ E 0 ( d z ^ ) = e i k d ( a d ) 3 ( 1 i k d k 2 d 2 ) α ˜ x ^
p in = γ e α ˜ x ^ ,
m in = 4 π a 3 β ˜ H 0 ( d z ^ )
m in = γ m c n 0 β ˜ y ^ ,
E tot , ff ( r r ^ , d ) = e i k r 4 π ε m ε 0 r k 2 [ ( 1 + γ e α ˜ exp i k d cos θ z ) sin ( θ x ) θ ^ x γ m β ˜ sin ( θ y ) exp ( i k d cos θ z ) φ ^ y ]
H tot , ff ( r r ^ , d ) = e i k r 4 π n 0 r c k 2 [ ( 1 + γ e α ˜ e i k d cos θ z ) sin ( θ x ) φ ^ x + γ m β ˜ sin ( θ y ) e i k d cos θ z θ ^ y ] .
P ( x , y , z ) = 1 2 ( E * ff × H ff ) = ω k 3 32 π 2 ε 0 ε m r 2 [ | 1 + γ e α ˜ e i k d cos θ z ) | 2 sin 2 ( θ x ) θ ^ x × φ ^ x + | γ m β ˜ | 2 sin 2 ( θ y ) θ ^ y × φ ^ y + ( 1 + γ e α ˜ e i k d cos θ z ) * sin ( θ x ) γ m β ˜ sin ( θ y ) e i k d cos θ z θ ^ x × θ ^ y + ( γ m β ˜ e i k d cos θ z ) * sin ( θ y ) ( 1 + γ e α ˜ e i k d cos θ z ) sin ( θ x ) φ ^ x × φ ^ y
cos ( θ l ) = l , sin ( θ l ) = 1 l 2 , cos ( φ l ) = m / 1 l 2 , sin ( φ l ) = n / 1 l 2 , θ ^ x × θ ^ y = z 1 x 2 1 y 2 r ^ , φ ^ x × φ ^ y = z 1 x 2 1 y 2 r ^ ,
P ( x , y , z ) = ω k 3 32 π 2 r 2 ε 0 ε m { ( 1 x 2 ) | 1 + γ e α ˜ e i k d z | 2 + ( 1 y 2 ) | γ m β ˜ | 2 + 2 z [ γ m * β ˜ * e i k d z ( 1 + γ e α ˜ e i k d z ) ] } r ^ .
P ( 0 , 0 , r ) ω 32 π 2 ε m ε 0 r 2 k 3 ( | γ m β ˜ | 2 + | 1 + γ e α ˜ e i k d | 2 2 | γ m β ˜ | | 1 + γ e α ˜ e i k d | ) r ^
arg ( e i k d + γ e α ˜ ) = arg ( γ m β ˜ ) ,
e i k d + γ e α ˜ = γ m β ˜ P ( 0 , 0 , r ) = 0
e i k d + γ e α ˜ γ m β ˜ ,

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